Average Rate of Change
Introduction:
The average rate of change of a function 𝑦 = 𝑓( 𝑥 ) measures how the output of the function changes, on average, as the input changes over a specified interval. It quantifies the "speed" of change in the output with respect to the input and is conceptually similar to the slope of a straight line connecting two points on the graph of the function.
Definition
The average rate of change of a function \(y = f(x)\) measures how the output of the function changes, on average, as the input changes over a specified interval. It is defined as:
Where:
- \(x_1, x_2\): Two points in the domain of \(f(x)\).
- \(f(x_1), f(x_2)\): are the corresponding function values at these points.
- The numerator, \(𝑓(𝑥_2)−𝑓(𝑥_1) \), represents the change in the function's output (vertical change).
- The denominator, \(𝑥_2 − 𝑥 _1\) , represents the change in the input (horizontal change).
The average rate of change is essentially the slope of the secant line that passes through the two points \((𝑥_1,𝑓(𝑥_1))\) and \((𝑥_2,𝑓(𝑥_2))\) on the graph of \(𝑓(𝑥)\).
Interpretation:
- A positive average rate of change indicates the function is increasing, on average, over the interval.
- A negative average rate of change indicates the function is decreasing, on average, over the interval.
- A zero average rate of change indicates that the function's output remains constant, on average, over the interval.
Examples
Example 1: Linear Function
Consider \[f(x) = 2x + 3 \quad over \quad [1, 4]\]:
- \[f(1) = 5, \qquad f(4) = 11\]
- \[ \text{Average Rate of Change} = \frac{f(4) - f(1)}{4 - 1} = \frac{11 - 5}{3} = 2 \]
The average rate of change is \(2\), equal to the slope of the line.
Example 2: Quadratic Function
Consider \[f(x) = x^2 - 3x + 2, \quad over \quad [1, 3]\]
- \[f(1) = 0, \qquad f(3) = 2\]
- \[ \text{Average Rate of Change} = \frac{f(3) - f(1)}{3 - 1} = \frac{2 - 0}{2} = 1 \]
The secant line has a slope of \(1\).
Example 3: Exponential Function
Consider \[f(x) = 2^x, \quad over \quad [0, 2]\]:
- \[f(0) = 1, \qquad f(2) = 4\]
- \[ \text{Average Rate of Change} = \frac{f(2) - f(0)}{2 - 0} = \frac{4 - 1}{2} = 1.5 \]
The average rate of change is \(1.5\). This value reflects the average growth rate of the exponential function between 𝑥 = 0 and 𝑥 = 2.
Summary
The average rate of change provides a useful measure of the overall behavior of a function over a specific interval. Unlike the instantaneous rate of change (i.e., the derivative), it does not provide detailed information about the function's behavior at every point in the interval. However, it is a simple and effective tool for understanding trends in the function's output.